Simulation of rare events, such as small failure probabilities, is a common problem in several scientific and engineering fields. This paper presents a novel Monte-Carlo-based simulation approach for this purpose, called the Accelerated Weight Histogram (AWH) method. The method was originally developed to solve challenging sampling problems in statistical and biological physics, but its algorithm has here been reformulated for estimation of rare event probabilities. The applicability of the method is investigated for a couple of simpler computational examples and for a more advanced practical case, which consists of a rock tunnel stability problem. To estimate the probability of failure of the latter in a realistic manner, a boolean indicator function was used to describe failure, based on a mechanical concept known as unbalanced force ratio. The investigated cases indicate that the AWH method performs well for both simpler limit states and more complex failure definitions.